I've got a very simple question yet it has been bugging me for ages, WHY/HOW does 0 permutations/arrangements =1 (0!=1)?

is the number of permutations of the set , i.e. the number of maps from to itself that are bijective.

If , then , and there exists one map from to ... the empty map !!

Remember how maps are defined: a map from to is a subset of such that, for every , there exists exactly one such that , and we write .

If , then is a subset of (whatever is) that satisfies the assumption since there is no (hence the condition is automatically fulfilled: it is empty). Thus, is a map from to anything. And it is the only one.

It is bijective from to : for every , there is a unique mapped to . Indeed, there is no such , so there's nothing to be checked.

Thus, is the only map from to itself, and it is bijective. As a conclusion, .